2022春高数下期末试题(回忆版)

一

Multiple Choice Questions: (only one correct answer for each of the following questions.)

1-1

The interval of convergence for the power series $\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n{3}^n}$ is

(A) $[-\frac{1}{3},\frac{1}{3}]$

(B) $[-\frac{1}{3},\frac{1}{3})$

(C) $[-3,3]$

(D) $[-3,3)$

1-2

Let $f(x,y)=\{\begin{array}{ll}{y}^2\sin \frac{1}{x^2+y^2},& (x,y)\ne (0,0)\\ 0,& (x,y)=(0,0)\end{array}$ Which of the following statement is wrong

(A) $f(x,y)$ is continuous at $(0,0)$

(B) $f_x(0,0)$ exists

(C) $f_y(0,0)$ exists

(D) $f_x(x,y)$ is continuous at $(0,0)$

1-3

If $f(x,y)$ has partial derivatives at $(x_0,y_0)$, then

(A) $f(x,y)$ is bounded around $(x_0,y_0)$

(B) $f(x,y)$ is continuous around $(x_0,y_0)$

(C) $f(x,y_0)$ is continuous at $x_0,f(x_0,y)$ is continuous at $y_0$

(D) $f(x,y)$ is continuous at $(x_0,y_0)$

1-4

Let $a$ be a constant, Then the series $\sum _{n=1}^{\mathrm{\infty }}(\frac{\sin (an)}{n^2}+\frac{(-1{)}^n}{n+1})$

(A) converges absolutely

(B) converges conditionally

(C) diverges

(D) the convergence depends on the value of $a$

1-5

${\int }_0^1{\int }_y^1\frac{\cos x}{x}dxdy=$

(A) cos 1

(B) $\sin 1$

(C) $1-\cos 1$

(D) $1-\sin 1$

二 Please fill in the blank for the questions below

2-1

If the function $z=z(x,y)$ is determined by $x^2-2y^2+z^2-4x+2z-5=0$, then ${\frac{\partial z}{\partial y}|}_{(5,2,2)}=$ $\underset{―}{}$

2-2

$\underset{(x,y)\to (0,0)}{lim}\frac{\sin \left(x{y}^2\right)}{x^2+y^2}=$ $\underset{―}{}$

2-3

If the region $D=\{(x,y)\mid x^2+y^2\le 1\}$, then ${\iint }_D{e}^{-x^2-y^2}dxdy=$ $\underset{―}{}$

2-4

Let $F=(z+e^{\sin y})i+(\cos z-y)j+(2z+\ln (1+y^2))k$ .D is the upper semi-sphere $0\le z\le \sqrt{a^2-x^2-y^2}$ $(a\ge 0)$, and $S$ is the boundary of the region $D$ .Then the outward flux of $F$ across $S$ ; ${\iint }_{s}F\cdot nd\sigma =$ $\underset{―}{}$

(5) ${\int }_{c}yz{e}^{xz}dx+e^{xz}dy+xy{e}^{xz}dz=$ $\underset{―}{}$, where $C$ is a path from $(2,1,0)$ to $(0,4,5)$

三

Find the equation for the plane through the origin parallel to the following lines:

$$l_1=\{\begin{array}{l}x=1\\ y=-1+t\\ z=2+t\end{array}$$

,

$$l_2=\{\begin{array}{l}x=-1+t\\ y=-2+2t\\ z=1+t\end{array}$$

四

Use Taylor series to evaluate $\underset{n\to \mathrm{\infty }}{lim}(n^3\sin \frac{2}{n}-2n^2)$

五

In what directions is the directional derivative of $f(x,y)=xy+y^2$ at $P(3,2)$ equal to zero?

六

Compute ${\iint }_{D}xydxdy$, here $D$ is the disk enclosed by the curve $x^2+y^2=2x+2y$

(Hint=use substitution)

七

Find the centroid of the region $D=\{(x,y,z)\mid \sqrt{x^2+y^2}\le z\le \sqrt{1-x^2-y^2}\}$

八

Calculate the line integral ${\oint }_{c}F\cdot dr$, where $F=(y^2+e^{e^x})i+(xy+\cos y)j+xzk$, and $C$ is the curve of intersection of the cylinder $x^2+y^2=4y$ and the plane $y=z$ courterdockwise when viewed from above

九

Find the absolute maximum and minimum values of the function $f(x,y)=3x^2+4xy$ on the region $R:x^2+y^2\le 1$